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Penjalaran Gelombang Seismik di dalam medium elastik-isotropik

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Some facts about the contributions of seismic method in earth sciences and exploration geophysics:

The knowledge of the interior of the earth comes from earthquake seismic method

Has contributed in proving the Plate Tectonic Theory.

Has succeeded used in mapping depth of bedrock, crustal thickness, and uppermost mantle velocity, especially by using seismic refraction method.

But,

Like other geophysical methods, the seismic method is only an indirect method: Only the physical property of solid matter (elastic constants, velocities) can be mapped.

The most important contribution is in oil industries: can show details of layering within sedimentary basins and gross structure of the deeper crust by using seismic reflection method.

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Seismic Waves - Definitions

Seismic:Relating to an earthquake or artificial shaking of the earth.

Waves:Motion that periodically advances and retreats as it is transmitted Motion that periodically advances and retreats as it is transmitted progressively from one particle in a medium to the next.

Seismic wave:propagation of energy through the Earth caused by earthquake or artificial vibrations.

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Elastic:ability of a material to return immediately to its original size, shape, or position after being squeezed, stretched, or otherwise deformed.

Elasticity

Undeformed

DeformedIn seismic method:

Stress

∆L

L

Deformedwhenstressed

Return tooriginal shapewhen stressremoved

In seismic method:Earth’s materials must behave elastically in order to transmit a seismic wave.The degree of elasticity thus determines how well a material transmits a seismic wave.

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Elasticity

Earth materials may beElastic, under some conditions, OR

Inelastic, under others.

the magnitude and orientation of the deforming stress (amount of compression, tension, or shearing).

Str

ess Ductile

It depends on:

tension, or shearing).

the length of time the material takes to achieve a certain amount of distortion (strain rate).

Strain

Strain = ∆∆∆∆L/LElastic limit

e.g. behavior of asthenosphere

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Apakah ada material In-elastic?

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Hooke’s Law

The linear relation between stress and strain is formulated by Hooke’s Law.

σij = Cijkl εkl

Cijkl = stiffness tensor or tensor of elastic parameters (elastic tensor)

σσσσij = stress tensor, describing the stress condition at any point x in medium

εεεεkl = strain tensor.εεεεkl = strain tensor.

The tensor Cijkl is 4th

order tensor and has originally 81 elastic components

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Elastic-Anisotropic and Elastic-Isotropic MediumBecause stress and strain are symmetric tensors:

σσσσij = σσσσji Cijkl = Cjikl, and

εεεεkl = εεεεlk Cijkl = Cijlk

The number of elastic tensor components reduces to 36 !!!With additional assumption related to thermodynamic (Cijkl = Cklij), the With additional assumption related to thermodynamic (Cijkl = Cklij), the number of elastic tensor reduces to 21 !!!

This material is called triclinic or elastic-anisotropic medium.

Materials which are completely symmetric: elastic-isotropic materials,which have only 2 independent elastic tensor components:Lame’s parameter λ and µ.

Cijkl = λ δλ δλ δλ δij δδδδkl + µµµµ ( δδδδik δδδδjl + δ+ δ+ δ+ δil δδδδjk ))))

σσσσij = λ δλ δλ δλ δij εεεεll + 2µµµµ εεεεij

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Bulk modulus or incompressibility: k

Elastic Constantsand their relationships with stress-strain

(describe the ability of a material to resist being compressed)

VVP

strainstress

k/∆

∆==

∆V ≅ 0 incompressible (k = ∼)

P

V

P'

V'

∆V

a b

∆P = P’ - P = pressure change

(applied stress)

∆V = V - V’= change in volume

caused by ∆P

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Shear modulus or rigidity or Lame’s constant µ:(describe the ability of a material to resist shearing)

llAF

strainstress

//

∆∆==µ

∆l ≅ 0 very rigid (µ = ∼)∆l ≅ ∼ No resistance to shearing (lack rigidity, µ = 0),

e.g. water.

Elastic Constantsand their relationships with stress-strain

e.g. water.

A

∆F

∆∆∆∆l

a b

l l

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Young modulus or stretch modulus: E

Elastic Constantsand their relationships with stress-strain

(describe the behavior of a material that is pulled or compressed)

a L

LLAF

strainstress

E/

/∆

==

Poisson’s ratio: ν

b

WA

W∆∆∆∆W

∆∆∆∆L

F

L(for a stretched rod: the ratio of transverse strainto longitudinal strain)

LLWW

//

∆∆=ν

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Lame’s constant λ:(illustrates the relationship between the four constants discussed above)

Elastic Constantsand their relationships with stress-strain

2 νµλ =−= Ek

)21)(1(3 ννλ

−+=−= k

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Elastic Constantsand their relationships with velocities

Cijkl = λ δλ δλ δλ δij δδδδkl + µµµµ ( δδδδik δδδδjl + δ+ δ+ δ+ δil δδδδjk ))))

C1111 = λ + 2 µ = λ + 2 µ = λ + 2 µ = λ + 2 µ = Vp2222 . ρ. ρ. ρ. ρ

µC

ρµ

ρµλ

ρ3/421111 +=+== kC

vp

C1212 = µ = µ = µ = µ = Vs2222 . ρ. ρ. ρ. ρ

ρµ

ρ== 1212C

vs

Thus, there are several types seismic waves:

Body waves: - a compressional wave, or a primary wave, or a P-wave, or a longitudinal wave, or a push-pull wave.

Surface waves: Rayleigh wave, Love wave, Stoneley wave.

- a shear wave, or a secondary wave or S-wave, ora transversal wave.

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Seismic Waves

a

b

Direction of Propagation

P

SVSH

Compressional Wave

Shear Wave

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Rock Properties

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Equation of motion

,3,2,1,,2

2

=∂∂

+=∂∂

jix

ftu

j

iji

iσ

ρ

u = u(x,t) = 3D ground displacement vector.x = [x1,x2,x3]T is a shorthand notation for cartesian coordinates.ρ = density of medium.f = components of the body force.

describe the conservation of momentum

fi = components of the body force.

Relationship between strain & displacement

��

�

�

��

�

�

∂∂

+∂∂=

i

j

j

iij x

u

xu

21ε

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Equation of Motion The complete equation of motion in the stress-velocity forms for 3D Elastic-Isotropic Medium:

yyzyyxyy f

zyxt

v+

∂∂

+∂

∂+

∂∂

=∂

∂ σσσρ

xxzxyxxx f

zyxtv +

∂∂+

∂∂

+∂

∂=∂

∂ σσσρ

v ∂∂∂∂ σσσy

v

zv

y

v

xv

tyzyxyy

∂∂

+���

����

�

∂∂+

∂∂

+∂∂=

∂∂

µλσ

2

xv

zv

y

v

xv

txzyxxx

∂∂+��

�

����

�

∂∂+

∂∂

+∂∂=

∂∂ µλσ

2

zzzyzxzz f

zyxtv +

∂∂+

∂∂

+∂

∂=∂

∂ σσσρzv

zv

y

v

xv

tzzyxzz

∂∂+��

�

����

�

∂∂+

∂∂

+∂∂=

∂∂ µλσ

2

���

����

�

∂∂+

∂∂

=∂

∂yv

x

v

txyxy µ

σ

��

���

�

∂∂+

∂∂=

∂∂

zv

xv

txzxz µσ

���

����

�

∂∂

+∂∂=

∂∂

z

v

yv

tyzyz µ

σ:,,

:,,

:,,

yzxzxy

zzyyxx

zyx vvv

σσσσσσ

particle velocity component

normal stresses

shear stresses

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On an interface

Converted waves will be produced !!!

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Explosion point source, e.g. TNT.Single-Force point source, e.g. Hammer.Double-couple point source, e.g. earthquake.

X-Y & R-T components of receivers

Types of seismic point sources

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Explosive point source and single-force point source

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Double-couple, e.g. earthquake.

P-wave

S-wave

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Double-couple, e.g. earthquake.

P-wave

S-wave

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Some applications

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Single-force point source applied in homogeneous-isotropic layering medium(Schmidt-Aursch, 1998)

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Single-force point source applied in homogeneous-isotropic layering medium

Phase 1: direct P Phase 2: P-T_12-P Phase 3: P-R_12-P Phase 4: direct S

Phase 1: direct P Phase 2: P-R_23-P Phase 3: P-R_12-P Phase 4: direct S

Phase 5: P-TK_12-S Phase 6: P-RK_12-S

Phase 1: direct P Phase 2: P-R_23-P Phase 3: P-T_12-P Phase 4: P-T_12-P-R_O-P Phase 5: direct S Phase 6: S-R_23-S

Phase 7: S-T_12-S Phase 8: P-T_12-P-RK_O-S

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Single-force point source applied in homogeneous-isotropic layering medium

Phase 1: P-T_12-P Phase 2: S-TK_12-P Phase 3: P-TK_12-S Phase 4: S-T_12-S Phase 5: P-R_23-P-T_12-P Phase 6: P-T_12-P-R_O-P-R_12-P Phase 7: Multiple Phase 8: Multiple

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Earthquake in Subduction Zone(Schmidt-Aursch, 1998)

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Earthquake in Subduction Zone � Some snapshots

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Earthquake in Subduction Zone � The synthetic seismograms

Phase 1: P-T_OM-P Phase 2: P-TC_OM-S Phase 3: S-TC_OM-P Phase 4: S-T_OM-S Phase 5: P-T_OU-P plus others Phase 6: S-T_OU-S plus others

P: Longitudinal wave S: Transversal wave R: Reflection T: Transmission C: Conversion OU: at border between upper crust and lower crust OM: at border between upper crust and Mantel UM: at border between lower crust and Mantel

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Black ForestSule (2004)

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2D Elastic Waveform ModelingSP-21, homogeneous ore-dyke model, Explosion source

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2D Elastic Waveform ModelingSP-21, homogeneous ore-dyke model, Explosion source

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Simulasi Historical Gempa Jericho (1927)Gottschaemmer (2002)

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Pembuatan model elastic-isotropic, snapshots, dan synthetic seismograms

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Peak Ground Velocity

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Studi Kasus 2: Supervolcano Toba

Model Vp

Sule (2006), model kecepatan diambil dari Wandono et. al. (2006)

Model Vs

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Simulasi perambatan gelombang seismik (Sule, 2006)

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References:

1. Aki, K. and Richards, P. G., (1980), Quantitative Seismology, W.H. Freeman and Company.

2.Lillie, R. L., (2002), Whole Earth Geophysics: An Introductory Textbook for Geologists and Geophysicists, Pearson International.

3. Schmidt-Aursch, M., (1998), Zweidimensionalle Modellierung von Wellenausbreitung bei erdbeben mit finiten differenzen, von Wellenausbreitung bei erdbeben mit finiten differenzen, Master’s thesis, Universitaet Karlsruhe (TH).

4. Sule, R., (2004), Seismic Travel Time Tomography and Elastic Waveform Modeling - Application to Ore-Dyke Characterization, Logos Verlag Berlin.